Thursday, September 14, 2023

The Locker Problem

 

First, I like to visualize word problems and draw them out. I started with small steps and drew 10 lockers and 10 students. I used an x to represent a closed locker and a checkmark to represent an open locker. As I got to student #10, I realized lockers #1, 4, and 9 were closed and the rest were open. Hence, I noticed a pattern and extended my strategy to 1,000 lockers. I was able to come to the conclusion that locker numbers that are perfect squares are closed and the rest are open.


Updated November 7, 2023
When the state of the locker changes, it represents the factor of that locker number. This will be clearly seen when highlighting where the locker changes its state. For example, locker 4 will change states when it is students #1, 2, and 4's turn. Hence, when we think about the locker's factors, we can conclude that the locker numbers that are perfect squares are closed.

Updated November 8, 2023
The square numbers have an odd number of factors because the square root of a perfect square does not have a pair. For example, the factors of 9 are 1,3,9 and it has an odd number of factors because the square root of 9 does not have a pair. 
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2 comments:

  1. Susan GerofskyNovember 5, 2023 at 8:47 AM

    OK -- good diagram and good first step (recognizing a potential pattern). But you have not talked about the second step that is also very important: exploring WHY the locker numbers that are perfect squares are the closed ones! This is in the nature of a proof, though not necessarily a formal one. Please complete your explanation, and let me know when you have done so, so that I can mark this post as complete.

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  2. Susan GerofskyNovember 7, 2023 at 5:10 PM

    OK, I'll accept this, but I would really like it if you mentioned why it is that square numbers have an odd number of distinct factors, while every other kind of number (including primes) has an even number of factors!

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